Perturbation resilience for the facility location problem

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Discrete Mathematics and Mathematical Programming

MASTER’S THESIS

Perturbation resilience for the facility location problem

M.B. Tijink

Supervisor:

Dr. B. Manthey (University of Twente, DMMP) Graduation committee:

Prof.dr. M.J. Uetz (University of Twente, DMMP) Dr. B. Manthey (University of Twente, DMMP) Dr. J.C.W. van Ommeren (University of Twente, SOR)

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Abstract

Several NP-hard problems, e.g. 𝑘-means clustering, are solved very quickly and near optimality for practical instances, even though there are known worst-cases from a theoretical perspective.

These worst-cases do not seem to appear in practice. There even is a saying; “Clustering is either easy or pointless”, indicating that the difficulty of these problems is only a theoretical concern.

Several approaches to formalize this saying, by assuming that instances have a natural stability property in practice, recently proved to be successful in closing the gap between theory and practice.

In this thesis we look at the facility location problem, which is NP-hard. We add the 𝛾-perturbation resilience assumption, which requires that the instance must allow small perturbations of all costs in the instance without changing its optimal solution. Many instances in practice likely already satisfy this assumption, so we focus on the theoretical impact of this assumption. We found several consequences this assumption; local search algorithms will always result in the optimal solution, for 𝛾 ≥ 3, and may not give the optimal solution for all 𝛾 < 3. We also show that several greedy algorithms do not work either to solve the facility location problem with the 𝛾-perturbation resilience assumption, even for high 𝛾. A relation between this assumption and approximation algorithms seems obvious, but we prove that this relation is false. Finally, we show that, for small 𝛾, the existence of an efficient algorithm to solve the facility location problem with the 𝛾-perturbation resilience assumption implies that RP = NP.

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1 Introduction

In this thesis we look at the facility location problem. This is a hard problem, as further explained in subsection 1.1, but the theoretical results for this problem show it to be more difficult than is observed in practice [17]. To solve this contrast between the theoretical and practical difficulty, we introduce the 𝛾-perturbation resilience assumption for the facility location problem (see subsection 1.2): 𝛾-perturbation resilient instances of the facility location problem do not have different optimal solutions when perturbed with small amounts. This assumption is somewhat natural: we think that many instances already have this property in practice, yet it makes the problem easier from the theoretical perspective. We will look into the consequences of making this assumption. In the following two subsections we will go into more detail about the facility location problem and the difference between NP-hardness in theory and practice, and we will detail the contributions made in this thesis in the final subsection.

1.1 The facility location problem

In the facility location problem we have a finite set of customers and a finite set of locations where we can open a facility. The goal of the facility location problem is to serve all customers while minimizing the total cost. This requires that each customer is connected to exactly one facility, which must be open. Thus, the key choice is which facilities to open and the secondary choice is which customer connects to which open facility. For every customer-location pair we have an associated cost, which we must pay if we we open a facility on that location and let the facility serve that customer. We also have associated costs for every location, which we must pay if we open a facility at that location. Thus, there is a tradeoff between opening many facilities such that the customer-location costs stay low and opening few facilities such that the facility costs are low. A common application of the facility location problem is determining where to build e.g. warehouses or hospitals. For more details on the facility location problem, see Krarup and Pruzan [20] or Melo et al. [25].

A lot of variants of this simple problem exist. For example, the uncapacitated facility location problem, as described above, allows an arbitrary number of customers to connect to an open facility. Thus, facilities must be very flexible, but this is not realistic for all application. The capacitated facility location problem limits the number of customers to connect to an open facility.

This way you could, for example, limit a warehouse capacity to five customers. If the capacity is reached, another facility needs to be opened to serve the remaining customers, either at the same location or at another location. The metric facility location problem requires the customer- location costs to be metric, making them behave like distances. This variant is suitable when the distance or time used in a road network is important. The non-metric facility problem does not restrict the customer-location costs, making it more suited for optimizing other types of resources. Some facility location problem variants have per-customers demands and per-unit customer-location prices, instead of just a single customer-location cost. This may not seem useful on its own, but is useful when combined with e.g. the metric facility location problem, as the demand and unit price formulation of the facility location problem allows more instances when these prices must be metric. And, of course, these variants can usually be combined for even more variants of the facility location problem [25, 32].

The facility location problem can also be seen as a variant of the 𝑘-means problem, but with variable amount of cluster centers to open and costs depending on which cluster center you open [11, 12, 24].

Even the simplest of the facility location problem variants – the uncapacitated metric facility location problem, which we use in this thesis – is NP-hard, which implies there is no proof yet that deterministic polynomial time algorithms exist for solving all possible instances [20, 25].

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The uncapacitated non-metric facility location problem is even NP-hard to approximate within a factor 𝑂(log |𝐷|), where |𝐷| is the number of customers [28, 32], so even guaranteeing a good solution is a hard problem.

The uncapacitated metric facility location problem, which we use in this thesis, is NP-hard to approximate within a factor 1.463 [16]. On the other hand, there are approximation algorithms that guarantee a solution within a factor 1.5 of the optimal solution [10].

1.2 NP-hardness in practice

Problems in computer science, e.g. the shortest path problem or the traveling salesman problem, can be classified depending on how fast they can be solved, for the fastest algorithm which solves them. So even if a problem A has a very slow algorithm, there could also be faster algorithms, so A does not need to be classified corresponding with the slow algorithm. The most well known complexity classes are P, meaning Polynomial, and NP, meaning Nondeterministic Polynomial.

If a problem A is in P, there is an algorithm that solves A in polynomial time, for any possible instance of A. Problems in NP can be solved in non-deterministic polynomial time or, equivalently, solutions to problems in NP can be verified in polynomial time given a proof of the solution [15, 27]. We often say a problem A in P is easy or that it has an efficient algorithm. On the other hand, we call call a problem A hard or say that it has only inefficient known algorithms if A is in NP. Other complexity classes also exist, for example we use complexity class RP, meaning Randomized Polynomial, in this thesis. Algorithms for problems in RP can use random numbers, so their output is not necessarily deterministic. An algorithm for some problem A in RP must always answer no if the given instance of A has no as answer, and the algorithm must answer yes with probability at least ¹₂ if the answer is yes. [27].

Proving that a problem is at least as hard as other problems is usually done with reductions. A reduction transforms instances of some problem B to an instance of some problem A such that the answer to the instance of problem B is identical to the answer of transformed instance. This way, if you have algorithm which can solve A, you can solve B by first running the transformation and then solving the resulting instance of problem A. Thus, a reduction proves that problem B is at least as hard as problem A [15, 27].

Cook showed that every problem in NP can be reduced to the boolean satisfiability problem [13], making it the first so-called NP-complete problem. If you prove that an NP-complete problem is in P, you have proved that P = NP. Likewise if you prove that an NP-complete problem is not in P, you have proved that P ≠ NP. Thus, NP-complete problems can be seen as a characterization of NP. Using a reduction from an NP-complete problem B to some other problem A in NP proves that A also is NP-complete. Examples of NP-complete problems are the SAT, clique, traveling salesman and knapsack problems [19]. The facility location problem is also NP-complete [20].

Despite all the research being done on those problems, to date no algorithms have been found to solve these problems in deterministic polynomial time; the question “P = NP or P ≠ NP?” is still open. For optimization problems, where the goal is to find the optimal solution, this is even worse: some problems of approximation, where the goal is to find a solution with value within a certain factor of the optimal solution, are still NP-hard, e.g. for the facility location problem and traveling salesman problem [14, 16].

In practice these theoretical problems have turned out to be less important: heuristics and local search algorithms often yield good results, even though they have worst cases on which they do not give good results or take exponential time [30]. This even gave rise to the saying “Clustering is either easy or pointless” [8]. Further examples are 2-OPT for the traveling salesman problem [18]

and the simplex algorithm [30]. It seems as if those difficult, worst-case instances do not appear in practice, either because these instances are very rare or because practical instances have some

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1. Introduction

inherent properties making them easier. The first of those reasons gave rise to smoothed analysis, in which small random perturbations are applied to instances. The idea in smoothed analysis is that the difficult instances are so specific that nearly all similar instances are not very difficult.

This has been successfully applied to e.g. the simplex algorithm [30] and the 2-OPT heuristic for the traveling salesman problem [22].

Another approach to better understand and analyze NP-hard problems is to constrain which instances are allowed. Examples of this approach are looking only at planar graphs instead of all graphs, or looking at the traveling salesman problem with metric distances. This yields useful results, but is generally not applicable in practice since real-world instances do not have those properties. A more recent approach is to choose a assumption on the allowed instances which is true, close to true or often true in practice [9]. An example where this approach was used is for MAX-CUT, where instances which are stable under perturbation can be solved efficiently [8].

Graph coloring is efficiently solvable if you assume that adding a couple of edges at arbitrary places in the graph does not increase its chromatic number [21]. The traveling salesman problem becomes easier if its optimal solution is some factor better than non-optimal solutions [26]. The most research with these kinds of assumptions has been done on the (𝑘-means) clustering problem, with several approaches [1, 4, 5, 6, 7, 29].

In the 𝑘-means clustering problem you have a set of points which you have to divide into 𝑘 clusters. Equivalently, you can choose 𝑘 cluster centers, as these directly correspond to clusters, and vice versa. The goal is to find clusters that minimize the sum of distances from point to its assigned cluster center. Awasthi et al. looked at 𝛾-perturbation stability for clustering: if you assume that the 𝑘-means instance has the same optimal solution after arbitrary perturbations of at most a factor 𝛾, then finding this solution is relatively easy [4].

As the facility location problem can be seen as a variant of the 𝑘-means clustering problem, we will use a 𝛾-perturbation resilience assumption similar to Awasthi et al. in this thesis. It seems likely that many real-world instances already satisfy this 𝛾-perturbation resilience, for small 𝛾. In this thesis we will see how the 𝛾-perturbation resilience assumption makes the facility location problem easier.

1.3 Contributions made in this thesis

In this thesis we first look at the facility location problem in general, before adding the 𝛾-perturbation resilience assumption and seeing what basic consequences this has (Section 2).

Then we look at several reasons why the facility location problem becomes somewhat easier with 𝛾-perturbation resilience. The first such reason is that local search algorithms will always find the optimal solution for 𝛾-perturbation resilient instances with 𝛾 ≥ 3. On the flip side, there exist (3 − 𝜀)-perturbation resilient instances that have local minima not equal to the optimal solution (Section 3). Several greedy algorithms have similar problems for small 𝛾, as we will show: these greedy algorithms make mistakes while determining the solution, even for 𝛾-perturbation resilient instances with relatively high 𝛾 (Section 4). This turns out to happen even for one of the currently best performing approximation algorithms, the Jain-Mahdian-Saberi algorithm [17]. This is a surprise, since a connection between 𝛾-perturbation resilience and approximation algorithms seems obvious. We prove that this connection is unfounded with a counterexample (Section 5).

We also look at how hard the facility location problem is with 𝛾-perturbation resilience, and found that if the ability to solve the perturbation resilient facility location problem for small 𝛾 exists, it implies that RP = NP (Section 6).

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2. The facility location problem

2 The facility location problem

The facility location problem has several variants. The variant we use in this thesis is usually called the uncapacitated metric facility location problem:

Definition. The uncapacitated metric facility location problem (referred to as the facility location problem or the FLP in this thesis) is the following optimization problem.

Instance: an instance (𝐹, 𝐷, 𝑓, 𝑐) of the FLP consists of a finite set of locations 𝐹, a finite set of customers 𝐷, facility costs 𝑓𝑖≥ 0 for all 𝑖 ∈ 𝐹 and service costs 𝑐𝑖𝑗≥ 0 for all 𝑖 ∈ 𝐹 , 𝑗 ∈ 𝐷.

The service costs are metric, i.e. 𝑐𝑖𝑗≤ 𝑐_𝑖^′_𝑗+ 𝑐_𝑖^′_𝑗^′+ 𝑐_𝑖𝑗^′ for all 𝑖, 𝑖^′∈ 𝐹 , 𝑗, 𝑗^′∈ 𝐷.

Solutions: a solution (𝑋, 𝜎) for an instance (𝐹, 𝐷, 𝑓, 𝑐) of the FLP consists of a nonempty set of open facilities 𝑋 ⊆ 𝐹 and a customer assignment 𝜎 ∶ 𝐷 → 𝑋 to open facilities.

Objective: The cost of a solution is 𝑐(𝑋, 𝜎) = ∑_𝑖∈𝑋𝑓_𝑖+∑_𝑗∈𝐷𝑐_{𝜎(𝑗)𝑗}. The objective is to minimize 𝑐(𝑋, 𝜎).

This variant is called uncapacitated since all facilities can handle an arbitrary number of customers and it is called metric because the service costs satisfy an extension of the triangle inequality.

The optimal solution to an FLP instance is usually denoted as (𝑋^∗, 𝜎^∗). The solution costs of the FLP, 𝑐(𝑋, 𝜎), can be split into two parts: the facility costs 𝑐𝐹(𝑋) = ∑_𝑖∈𝑋𝑓_𝑖and service costs 𝑐_𝑆(𝑋, 𝜎) = ∑_𝑗∈𝐷𝑐_{𝜎(𝑗)𝑗}. Additionally, given an instance of the FLP and a set of open facilities 𝑋, it is easy to compute an optimal corresponding customer assignment 𝜎: 𝜎(𝑗) = arg min_𝑖∈𝑋𝑐_𝑖𝑗, breaking ties arbitrarily. Thus, the customer assignment is often dropped in the cost notations, which implies that an optimal assignment is used.

Besides the metric FLP, as defined above, we sometimes use the non-metric version of the FLP in this thesis:

Definition. The uncapacitated facility location problem (referred to as the non-metric FLP in this thesis) is identical to the FLP except for one detail: an instance (𝐹 , 𝐷, 𝑓, 𝑐) of the non-metric FLP has service costs which need not be metric, i.e. 𝑐𝑖𝑗≰ 𝑐_𝑖^′_𝑗+ 𝑐_𝑖^′_𝑗^′+ 𝑐_𝑖𝑗^′ is allowed.

Thus, an instance of the FLP is also an instance of the non-metric FLP.

2.1 Results on the facility location problem

There is a decision variant of the FLP, where for a given instance (𝐹 , 𝐷, 𝑓, 𝑐) and bound 𝐿 the question is whether the optimal solution 𝑋^∗ has cost 𝑐(𝑋^∗) ≤ 𝐿. This decision variant of the FLP is NP-complete, as we will show in the following theorem and proof (for more information, see Krarup et al. [20]).

Theorem 1. The decision variant of the FLP is NP-complete.

Proof. First, we show that the decision variant of the FLP is in NP. For an arbitrary given instance (𝐹 , 𝐷, 𝑓, 𝑐) and bound 𝐿, choose an arbitrary nonempty set 𝑋 ⊆ 𝐹 nondeterministically.

Now check if the cost 𝑐(𝑋) ≤ 𝐿. If so, output “yes”, otherwise, output “no”.

If the instance has 𝑐(𝑋^∗) ≤ 𝐿, this will output “yes”, since 𝑋^∗ ⊆ 𝐹 and 𝑋^∗ is nonempty, so it will be one of the nondeterministic choices. If the instance has 𝑐(𝑋^∗) > 𝐿, then because 𝑐(𝑋) ≥ 𝑐(𝑋^∗) > 𝐿 no nondeterministic choice 𝑋 can yield “yes”, so the answer will be “no”.

Thus, the decision variant of the FLP is in NP.

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Now, to show that the decision variant of the FLP is NP-hard, we reduce the set-covering problem to the FLP. Because the decision variant of the set-covering problem is NP-complete [19], this will prove that the decision variant of the FLP is NP-hard.

The instances (𝑈, 𝑆) of the set-covering problem consist of the universe 𝑈 and a collection of subsets 𝑆 = ⋃ 𝑆𝑖, where each 𝑆𝑖 is a subset of 𝑈. The goal of the set-covering problem is to find the minimum cardinality set 𝐶 ⊆ 𝑆 that covers 𝑈, i.e. {𝑥 ∣ 𝑥 ∈ 𝑆𝑖, 𝑆_𝑖∈ 𝐶for some 𝑖} = 𝑈.

Given an instance (𝑈, 𝑆) and bound 𝑘, construct the following FLP instance and bound:

𝐹 = 𝑆, 𝐷 = 𝑈 , 𝑓_𝑖= 1,

𝑐_𝑖𝑗= {𝑘 if 𝑗 ∈ 𝑖, and 3𝑘 otherwise, 𝐿 = 𝑘 + 𝑘|𝐷|.

Clearly, if (𝑈, 𝑆) is a yes-instance, so is (𝐹, 𝐷, 𝑓, 𝑐), by opening facilities 𝑋 = 𝐶, yielding 𝑐(𝑋) = |𝐶|+𝑘|𝐷| ≤ 𝑘+𝑘|𝐷| = 𝐿. If (𝐹 , 𝐷, 𝑓, 𝑐) is a yes-instance, some 𝑋 has 𝑐(𝑋) ≤ 𝐿 = 𝑘+𝑘|𝐷|, so 𝑐𝜎(𝑗)𝑗= 𝑘for every 𝑗 ∈ 𝐷. Thus, 𝐶 = 𝑋 is a set cover with |𝐶| = 𝑐(𝑋) − 𝑘|𝐷| ≤ 𝑘 and (𝑈, 𝑆) is a yes-instance. As a result, this is a valid reduction, proving that the decision variant of the FLP is NP-hard.

As a consequence of this theorem, the (search variant of the) FLP is NP-hard.

NP-hard optimization problems often still have approximation algorithms which run in deterministic polynomial time, which give a solution with cost at most a factor 𝛼 > 1 times the optimal cost. Depending on the problem, there may be a lower bound for 𝛼, below which the problem is still NP-hard. The following three theorems show the current best approximation hardness results for the FLP.

Theorem 2 (Guha and Kuller 1999 [16]). Approximating the FLP within a factor 𝛼 = 1.463 is NP-hard.

Theorem 3 (Byrka and Aardal 2010 [10]). The FLP can be approximated within a factor 𝛼 = 1.5by a deterministic polynomial time algorithm.

Theorem 4 (Li 2013 [23]). The FLP can be approximated with a expected factor 𝛼 = 1.488 by a randomized polynomial time algorithm, i.e. for any possible instance, the algorithm computes solutions 𝑋 such that 𝔼[𝑐(𝑋)] ≤ 1.488𝑐(𝑋^∗).

These results are not particularly promising if you want to find a good solution to your FLP instance, since solutions 50% more expensive than the optimal solution does not seem very attractive. However, in practice several of the best approximation algorithms do a lot better. For example, the Jain-Mahdian-Saberi algorithm, with an approximation ratio 𝛼 = 1.61, achieves a ratio of 1.03 on average on real-world instances [17]. This difference between the observed performance and guaranteed performance leads us to making an assumption about real-world instances, as we show in the next subsection.

2.2 Perturbation resilience

Seeing how the FLP is an NP-hard problem [20], it is only natural to look for conditions on the instances which make the problem easier. Based on ideas by Bilu et al. [8] and Balcan et al. [5], we want to make an assumption which many real-world instances already satisfy. The

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perturbation resilience assumption for clustering problems by Awasthi et al. [4], adapted to the FLP, looks like a good choice. First, we need to define what it means to perturb an FLP instance.

Definition. An instance (𝐹, 𝐷, 𝑓^′, 𝑐^′) of the non-metric FLP is a γ-perturbation of instance (𝐹 , 𝐷, 𝑓, 𝑐) of the FLP, with 𝛾 ≥ 1, iff 𝑓𝑖 ≤ 𝑓_𝑖^′ ≤ 𝛾𝑓_𝑖 for all 𝑖 ∈ 𝐹 and 𝑐𝑖𝑗 ≤ 𝑐^′_𝑖𝑗 ≤ 𝛾𝑐_𝑖𝑗 for all 𝑖 ∈ 𝐹 , 𝑗 ∈ 𝐷. If it is clear from the context which 𝐹 and 𝐷 are used, a 𝛾-perturbed instance can also be denoted as (𝑓^′, 𝑐^′).

A 𝛾-perturbed instance is any instance that is a 𝛾-perturbation of some FLP instance.

Note that a 𝛾-perturbed instance might or might not be metric. This definition only allows greater costs than the original instance, which might seem weird for a perturbation, but this is simply a matter of scaling. An FLP instance remains equivalent under scaling all costs with the same value, so an equivalent definition of 𝛾-perturbation also allowing lower costs would, for instance, be: 𝑓𝑖≤√

𝛾𝑓_𝑖^′≤ 𝛾𝑓_𝑖 and similar for the service costs.

Using this definition, we can define 𝛾-perturbation resilience for the FLP.

Definition. An instance (𝐹, 𝐷, 𝑓, 𝑐) of the FLP is γ-perturbation resilient with 𝛾 ≥ 1 iff all 𝛾-perturbations (𝑓^′, 𝑐^′)of (𝐹 , 𝐷, 𝑓, 𝑐) have the same unique optimal solution (𝑋^∗, 𝜎^∗).

For small 𝛾 (i.e. 𝛾 < 1.05 or similar), it seems likely that many real-world instances satisfy this assumption, since stability of solutions makes a lot of sense: if an instance is very sensitive to perturbations, then it means that the exact solution probably does not matter much. For small 𝛾, it also implies that the instance is resilient in the face of small changes to the costs. In practice, this often happens too: you are not going to change the location of your warehouse if it is 3%

farther away than previously estimated. Nevertheless, we will consider all values 𝛾 in this thesis.

In the conclusion (section 7) we will come back to the results and see if they are applicable in practice.

If costs of 𝛾-perturbed instances are compared, the notation of the perturbed costs follow from the names given to the perturbed instances. So, for example, if (𝑓^′, 𝑐^′)is a 𝛾-perturbed instance, then 𝑐^′(𝑋)denotes the cost of solution 𝑋 in the perturbed instance.

Any 𝛾-perturbation resilient instance of the FLP is a valid 𝛾-perturbation of itself, so 𝛾-perturbation resilience implies that the original instance has the same optimal solution as any of its perturbations.

Note that, since 𝛾-perturbation resilience requires an unique solution, there are instances of the FLP which are not 𝛾-perturbation resilient for any 𝛾, namely exactly those instances with multiple optimal solutions.

The definition of 𝛾-perturbation resilience, although intuitive, is somewhat hard to work with since it requires checking the optimal solution for all 𝛾-perturbations of an FLP instance. The following theorem introduces an equivalent but easier to check definition.

Theorem 5. Consider, for some instance (𝐹, 𝐷, 𝑓, 𝑐) of the FLP with optimal solution 𝑋^∗, customer assignment 𝜎^∗ and nonzero optimal cost 𝑐(𝑋^∗), the following 𝛾-perturbation:

𝑓_𝑖^′= {𝛾𝑓_𝑖 if 𝑖 ∈ 𝑋^∗, and 𝑓_𝑖 otherwise, 𝑐^′_𝑖𝑗= {𝛾𝑐_𝑖𝑗 if 𝜎^∗(𝑗) = 𝑖, and

𝑐_𝑖𝑗 otherwise.

This 𝛾-perturbed instance (𝐹, 𝐷, 𝑓^′, 𝑐^′)has the same optimal solution 𝑋^∗and customer assignment 𝜎^∗ iff the instance (𝐹, 𝐷, 𝑓, 𝑐) is 𝛾-perturbation resilient. If so, the solution (𝑋^∗, 𝜎^∗)is unique for both (𝐹, 𝐷, 𝑓, 𝑐) and (𝐹, 𝐷, 𝑓^′, 𝑐^′).

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Proof. If the instance is 𝛾-perturbation resilient, then the given 𝛾-perturbation has the same unique optimal solution, by definition of 𝛾-perturbation resilience.

Thus, assume that the solution (𝑋^∗, 𝜎^∗)is also the solution to the given (𝐹 , 𝐷, 𝑓^′, 𝑐^′)instance.

Consider all possible 𝛾-perturbations ( ̃𝑓, ̃𝑐)of the (𝐹, 𝐷, 𝑓, 𝑐) instance and construct the following 𝛾-perturbation:

̃𝑓^′_𝑖 = { ̃𝑓_𝑖 if 𝑖 ∈ 𝑋^∗, and 𝑓_𝑖 otherwise,

̃

𝑐^′_𝑖𝑗= {𝑐_𝑖𝑗̃ if 𝜎^∗(𝑗) = 𝑖, and 𝑐_𝑖𝑗 otherwise.

By construction, it holds that ̃𝑐(𝑋^∗, 𝜎^∗) = ̃𝑐^′(𝑋^∗, 𝜎^∗) and ̃𝑐^′(𝑋, 𝜎) ≤ ̃𝑐(𝑋, 𝜎) for all nonempty open facility sets 𝑋 ⊆ 𝐹 and customer assignments 𝜎 ∶ 𝐷 → 𝑋. Now relate 𝑐^′(𝑋^∗, 𝜎^∗)to ̃𝑐^′(𝑋^∗, 𝜎^∗) and 𝑐^′(𝑋, 𝜎) to ̃𝑐^′(𝑋, 𝜎):

𝑐^′(𝑋^∗, 𝜎^∗) = ̃𝑐^′(𝑋^∗, 𝜎^∗) + ∑

𝑖∈𝑋^∗

𝛾𝑓_𝑖− ̃𝑓_𝑖+ ∑

𝑗∈𝐷

𝛾𝑐_𝜎∗(𝑗)𝑗− ̃𝑐_𝜎∗(𝑗)𝑗, (1) 𝑐^′(𝑋, 𝜎) = ̃𝑐^′(𝑋, 𝜎) + ∑

𝑖∈𝑋∩𝑋^∗

𝛾𝑓_𝑖− ̃𝑓_𝑖+ ∑

𝑗∈𝐷 𝜎(𝑗)=𝜎^∗(𝑗)

𝛾𝑐_𝜎^∗_(𝑗)𝑗− ̃𝑐_𝜎^∗_(𝑗)𝑗. (2)

By assumption 𝑐^′(𝑋^∗, 𝜎^∗) < 𝑐^′(𝑋, 𝜎). So ̃𝑐^′(𝑋^∗, 𝜎^∗) < 𝑐^′̃(𝑋, 𝜎), because the sum terms in equation (1) are at most the sum terms in equation (2). As noted above, by construction it follows that ̃𝑐(𝑋^∗) < ̃𝑐(𝑋).

This holds for all 𝛾-perturbations ( ̃𝑓, ̃𝑐)of (𝐹 , 𝐷, 𝑓, 𝑐) and all solutions (𝑋, 𝜎). Because 𝑐(𝑋^∗) ≠ 0 the solution is unique, proving that the instance (𝐹, 𝐷, 𝑓, 𝑐) is 𝛾-perturbation resilient.

We now have two equivalent definitions for 𝛾-perturbation resilience, so it is useful to know for which range of 𝛾 𝛾-perturbation resilient instances exist. FLP instances need to be metric, so this is not a trivial fact. The following two theorems establish that 𝛾-perturbation resilient instances exist for all 𝛾 ≥ 1.

Theorem 6. If an FLP instance (𝐹, 𝐷, 𝑓, 𝑐) is 𝛾-perturbation resilient, it also is 𝛾^′-perturbation resilient for all 𝛾^′∈ ℝ which satisfy 1 ≤ 𝛾^′≤ 𝛾.

Proof. Let (𝐹 , 𝐷, 𝑓, 𝑐) be any possible 𝛾-perturbation resilient FLP instance, and 𝛾^′ ∈ ℝ be an arbitrary number which satisfies 1 ≤ 𝛾^′ ≤ 𝛾. Now consider all 𝛾^′ perturbations (𝑓^′, 𝑐^′) of (𝐹, 𝐷, 𝑓, 𝑐). Since 𝛾^′ ≤ 𝛾, (𝑓^′, 𝑐^′)is a valid 𝛾-perturbation of (𝐹 , 𝐷, 𝑓, 𝑐). By definition of 𝛾-perturbation resilience, this implies that the same solution is the unique optimal solution to both (𝐹 , 𝐷, 𝑓, 𝑐) and (𝑓^′, 𝑐^′).

This satisfies the requirement for 𝛾^′-perturbation resilience, proving the theorem.

Theorem 7. For all 𝑥 ∈ ℝ, there exist 𝛾-perturbation resilient instances of the FLP with 𝛾 > 𝑥.

Proof. Assume to the contrary that there is some 𝑥 such that all instances of the FLP are 𝛾-perturbation resilient with 𝛾 < 𝑥. Assume that 𝑥 > 1, which can be done without loss of

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generality. Now consider the following instance (see Figure 1):

𝐹 = {1_𝑓, 2_𝑓}, 𝐷 = {1_𝑑, 2_𝑑}, 𝑓_𝑖= 0,

𝑐_𝑖𝑗= {1 if 𝑖 = 𝑘𝑓, 𝑗 = 𝑘_𝑑 for some 𝑘, and 2𝑥 otherwise.

1_𝑓

2_𝑓

1_𝑑 2_𝑑

𝑐1𝑓1𝑑= 1 𝑐₁

𝑓2_𝑑 = 2𝑥

𝑐₂

𝑓1_𝑑 = 2𝑥 𝑐2𝑓2𝑑= 1

𝑓₁_𝑓= 0

𝑓₂_𝑓= 0

Figure 1: Example of Theorem 7 This instance has three possible solutions, with the following costs:

𝑐({1_𝑓}) = 2𝑥 + 1, 𝑐({2_𝑓}) = 2𝑥 + 1, 𝑐({1_𝑓, 2_𝑓}) = 2.

Since all 𝛾-perturbations increase the cost of a solution 𝑋 at most by a factor 𝛾, a trivial lower bound for the perturbation resilience 𝛾^′ of this instance can be calculated as follows:

𝛾^′=^2𝑥+1₂ = 𝑥 +¹₂. This is a contradiction to the assumption that all instances of the FLP are 𝛾-perturbation resilient with 𝛾 < 𝑥, since 𝛾^′> 𝑥.

For most NP-hard problems, it is trivial to find the optimal solution to an instance once you have an algorithm that solves the decision variant of the problem (i.e. “is the cost of the optimal solution below a given bound?”). The following theorem looks at the decision variant of the FLP for 𝛾-perturbation resilient instances.

Theorem 8. Assume a deterministic algorithm A exists that decides 𝑔(|𝐹|, |𝐷|)-perturbation resilient FLP instances (𝐹 , 𝐷, 𝑓, 𝑐) in polynomial time, for some function 𝑔. Then a deterministic algorithm exists that finds the optimal solution 𝑋^∗ for 𝛾^′-perturbation resilience FLP instances (𝐹 , 𝐷, 𝑓, 𝑐)with 𝛾^′> 𝑔(|𝐹 |, |𝐷|) in polynomial time.

This theorem implies that, like for many other NP-hard problems, discovering the value of a 𝛾-perturbation resilient instance is not more difficult than finding the corresponding solution. The only exception, where this may or may not be true, is the border case of exactly 𝑔(|𝐹 |, |𝐷|)-perturbation resilient instances. Although a theorem like this for general FLP can easily be proven using self-reducibility, this is not true here. Self-reducibility can destroy the 𝛾-perturbation resilience of an instance, and as result the assumed existing algorithm cannot be applied to the resulting instances.

Proof. Assume algorithm A exists for some function 𝑔. Now, take an arbitrary 𝛾^′-perturbation resilient instance (𝐹, 𝐷, 𝑓, 𝑐) with 𝛾^′> 𝑔(|𝐹 |, |𝐷|). First, calculate the exact optimal cost 𝑐(𝑋^∗) using algorithm A, which can be done in polynomial time using a binary search with calls to A.

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Because 𝛾^′> 𝑔(|𝐹 |, |𝐷|), it is possible to𝑔(|𝐹 |,|𝐷|)^𝛾^′ -perturb the instance (𝐹, 𝐷, 𝑓, 𝑐) to some instance (𝐹 , 𝐷, 𝑓^′, 𝑐^′). Because this is a valid 𝛾^′-perturbation, both instances yield the same optimal solution 𝑋^∗ and customer assignment 𝜎^∗. Note that all 𝑔(|𝐹 |, |𝐷|)-perturbations of instance (𝐹 , 𝐷, 𝑓^′, 𝑐^′)are valid 𝛾^′-perturbations of (𝐹, 𝐷, 𝑓, 𝑐). Thus, algorithm A can be used on all such instances (𝐹 , 𝐷, 𝑓^′, 𝑐^′), since they are 𝑔(|𝐹|, |𝐷|)-perturbation resilient, and use the same optimal solution 𝑋^∗ to decide if it is a yes- or no-instance.

So, consider the following 𝑔(|𝐹 |,|𝐷|)^𝛾^′ -perturbation, for some fixed facility 𝑖^′ ∈ 𝐹, which will be chosen later:

𝑓_𝑖^′= {

𝛾^′

𝑔(|𝐹 |,|𝐷|)𝑓𝑖 if 𝑖 = 𝑖^′, and

𝑓_𝑖 otherwise.

This perturbed instance (𝐹, 𝐷, 𝑓^′, 𝑐)has cost 𝑐^′(𝑋^∗) = 𝑐(𝑋^∗)iff 𝑖^′∉ 𝑋^∗. So by running algorithm A on instance (𝐹 , 𝐷, 𝑓^′, 𝑐)with threshold 𝑐(𝑋^∗), a part of the optimal solution 𝑋^∗ is discovered, if 𝑓𝑖^′= 0. If 𝑓𝑖^′= 0, we know that including 𝑖^′ in 𝑋^∗ never results in a higher cost. Doing this for all facilities 𝑖^′∈ 𝐹yields the optimal solution 𝑋^∗.

All parts of this algorithm run in polynomial time and use algorithm A a polynomial number of times, which completes the proof.

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3. Local search

3 Local search

Local search techniques are a common way to solve problems. In a local search algorithm, we always start from some initial solution, like a trivial solution (e.g. 𝑋 = 𝐹 for the FLP) or a solution computed by a heuristic. From there, local search tries to improve its current solution.

It does that by computing the neighbourhood of the current solution, a set of solutions which are similar to the current solution. What this neighbourhood is differs per problem and can be chosen by the creator of the local search algorithm. The local search algorithm now computes the cost of each solution in the neighbourhood. If at least one is better than the current solution, it updates its current solution to one of the better solutions. If none are, the local search algorithm terminates. All solutions which are at least as good as all their neighbouring solutions are called local minima.

Local search techniques are commonly used in practice, since they are easy to implement: they only require you to define a neighbourhood, which is quite simple. Additionally, real-world problems often have extra constraints on the solution, which do not correspond to the theoretical problems. An example for the FLP would be that there are a few facilities that, when open, must have exactly three assigned customers. These extra constraints are not difficult to add in a local search algorithm.

In this section we look at local minima and local search algorithms in general, and see how 𝛾-perturbation resilient instances impact them.

3.1 Existence of local minima

In this subsection we will show that 3-perturbation resilient instances do not have local minima except for the global minimum. We also show that there are (3−𝜀)-perturbation resilient instances for which this is not true, i.e. that there are local minima unequal to the optimal solution. To do so, we first define what a local minimum is for the FLP.

Definition. A solution 𝑋 to an instance of the FLP is a local minimum iff

• 𝑐(𝑋 ⧵ {𝑖}) ≥ 𝑐(𝑋) for all 𝑖 ∈ 𝑋 with |𝑋| ≥ 2 (dropping a facility), and

• 𝑐(𝑋 ∪ {𝑖}) ≥ 𝑐(𝑋) for all 𝑖 ∈ 𝐹 ⧵ 𝑋 (adding a facility), and

• 𝑐(𝑋 ⧵ {𝑖} ∪ {𝑗}) ≥ 𝑐(𝑋) for all 𝑖 ∈ 𝑋, 𝑗 ∈ 𝐹 ⧵ 𝑋 (swapping a facility).

This definition means that the local minimum 𝑋 cannot be improved by opening a new facility, closing an open facility or switching between an open and closed facility.

The following theorem is a result for local search algorithms on any FLP instance.

Theorem 9 (Arya et al. 2004 [3]). All local minima 𝑋 for an instance of the FLP satisfy 𝑐𝐹(𝑋) ≤ 𝑐𝐹(𝑋^∗) + 2𝑐𝑆(𝑋^∗)and 𝑐𝑆(𝑋) ≤ 𝑐𝐹(𝑋^∗) + 𝑐𝑆(𝑋^∗), where 𝑋^∗ is the optimal solution for the instance. Combined, this yields 𝑐(𝑋) ≤ 3𝑐(𝑋^∗).

This theorem will be used to show that there are no local minima except for the optimal solution for 3-perturbation resilient FLP instances (Theorem 11). To prove this, we first need an extra result from the following lemma. This lemma means that, if some 𝛾-perturbation resilient instance exists, then also such a 𝛾-perturbation resilient instance with more restrictions exists: you can see this as removing everything from an instance that is not needed to remain 𝛾-perturbation resilient or have a local minimum 𝑋 besides the optimal solution.

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Lemma 10. Assume a 𝛾-perturbation resilient instance (𝐹, 𝐷, 𝑓, 𝑐) of the FLP exists with a local minimum (𝑋, 𝜎) not equal to the optimal solution (𝑋^∗, 𝜎^∗).

Then an instance (𝐹^′, 𝐷^′, 𝑓^′, 𝑐^′)of the FLP exists with the following properties:

• the instance (𝐹^′, 𝐷^′, 𝑓^′, 𝑐^′)is 𝛾-perturbation resilient, and

• the instance (𝐹^′, 𝐷^′, 𝑓^′, 𝑐^′)has a local minimum (𝑋^′, 𝜎^′)not equal to the optimal solution (𝑋^′∗, 𝜎^′∗), and

• 𝐹^′= 𝑋^′∪ 𝑋^′∗, and

• 𝑓_𝑖^′= 0for all 𝑖 ∈ 𝑋^′∩ 𝑋^′∗, and

• for all 𝑗 ∈ 𝐷^′, (𝜎^′(𝑗) ∉ 𝑋^′∩ 𝑋^′∗ or 𝜎^′∗(𝑗) ∉ 𝑋^′∩ 𝑋^′∗).

Note that the optimal solution (𝑋^∗, 𝜎^∗)to any instance is always a local minimum, so this is explicitly excluded in the lemma.

Proof. Take any such instance (𝐹, 𝐷, 𝑓, 𝑐). We transform the instance (𝐹^′, 𝐷^′, 𝑓^′, 𝑐^′)using the following steps. We apply each step until its conditions are satisfied for all local minima 𝑋 ≠ 𝑋^∗, possibly applying earlier steps again in the process if their conditions are not valid any more after applying a later step. Thus, at the beginning of every step, the conditions of all previous steps are satisfied.

Step 1. Condition to satisfy: 𝐹 = 𝑋 ∪ 𝑋^∗.

Drop all facilities not in 𝑋 or 𝑋^∗, i.e. 𝐹^′= 𝑋 ∪ 𝑋^∗. All customer assignments of 𝜎 and 𝜎^∗ remain valid and the costs 𝑐(𝑋) and 𝑐(𝑋^∗)are unchanged. Thus, the resulting instance (𝐹^′, 𝐷, 𝑓, 𝑐)is still 𝛾-perturbation resilient with optimal solution 𝑋^∗. Because all subsets of 𝐹^′ also are a subset of 𝐹, 𝑋 ≠ 𝑋^∗is still a local minimum. After doing this, the conditions of step 1 are satisfied.

Step 2. Condition to satisfy: 𝑓𝑖= 0 for all 𝑖 ∈ 𝑋 ∩ 𝑋^∗. Change the facility costs to the following:

̃𝑓_𝑖= {0 if 𝑖 ∈ 𝑋 ∩ 𝑋^∗, and 𝑓_𝑖 otherwise.

Note that the resulting instance (𝐹, 𝐷, ̃𝑓, 𝑐) still has 𝑋 as a local minimum, since the cost of adding a facility to 𝑋 is identical, compared to instance (𝐹 , 𝐷, 𝑓, 𝑐), and the cost of dropping or swapping a facility from 𝑋 is the equal or higher, as compared to instance (𝐹 , 𝐷, 𝑓, 𝑐). To show that the instance (𝐹 , 𝐷, ̃𝑓, 𝑐) is 𝛾-perturbation resilient with optimal solution 𝑋^∗, consider all nonempty sets of open facilities 𝑌 ⊆ 𝐹 and customer assignments 𝜎^′ in all perturbations of costs 𝑓_𝑖^′ and 𝑐^′𝑖𝑗 and equivalent perturbations of ̃𝑓_𝑖 and 𝑐𝑖𝑗:

𝑐^′(𝑋^∗, 𝜎^∗) = 𝑐^′_𝐹(𝑋^∗) + 𝑐_𝑆^′(𝑋^∗, 𝜎^∗)

= ∑

𝑖∈𝑋^∗⧵𝑋

𝑓_𝑖^′+ ∑

𝑓_𝑖^′+ 𝑐^′_𝑆(𝑋^∗, 𝜎^∗)

= ̃𝑐^′(𝑋^∗, 𝜎^∗) + ∑

𝑓_𝑖^′, 𝑐^′(𝑌 , 𝜎^′) = 𝑐^′_𝐹(𝑌 ) + 𝑐_𝑆^′(𝑌 , 𝜎^′)

= ∑

𝑖∈𝑌⧵(𝑋∩𝑋^∗)

𝑓_𝑖^′+ ∑

𝑖∈𝑌 ∩𝑋∩𝑋^∗

𝑓_𝑖^′+ 𝑐^′_𝑆(𝑌 , 𝜎^′)

= ̃𝑐^′(𝑌 , 𝜎^′) + ∑

𝑖∈𝑌 ∩𝑋∩𝑋^∗

𝑓_𝑖^′.

Perturbation resilience for the facility location problem